$\def\AAA{\mathcal{A}} \def\sbe{\subseteq} \def\y{\psi} \def\w{\omega} \def\x{\chi} $
In section $1.26$ of Lee's Intro to Smooth Manifolds the concept of an open submanifold is introduced, but it is not shown that the defined atlas is smooth: "it is easy to verify that this is a smooth atlas for $U$":
Is the following a proper proof for the smoothness of $\AAA_U$?
Proof: The charts in $\AAA_U$ are smoothly compatible since $\AAA_U\sbe \AAA$, and it remains to be shown that $\AAA_U$ is maximal. If it wasn't there would be a chart $(V,\y)$ with $V\sbe U$ and smoothly compatible with every chart in $\AAA_U$, but not belonging to $\AAA_U$. We show $(V,\y)\in\AAA$, implying $\AAA$ is not maximal and causing a contradiction. Choose any chart $(W,\w)\in\AAA$. If $V\cap W = \emptyset$ we are done, otherwise we wish to show that $\w\circ\y^{-1}:\y(V\cap W)\to\w(V\cap W)$ is smooth. To do so choose $p\in V\cap W$ and a smooth chart $(X,\x)\in\AAA_U$ of $p$, so that $$\w\circ\y^{-1}=\w\circ\x^{-1}\circ\x\circ\y^{-1}:\y(V\cap W\cap X)\to \w(V\cap W\cap X).$$ In particular, we have that $\w\circ\y^{-1}$ is smooth at $p$. Since $p$ is arbitrary, we get that $\w\circ\y^{-1}$ is smooth. Thus $(V,\y)$ is smoothly compatible with any smooth chart in $\AAA$, but does not belong to $\AAA$, contradicting $\AAA$'s maximality.